448 IX- DYNAMICS OF DEFORMABLE BODIES. 



But since the three other sides are the projections of dS, we have 



85) dS x = dS cos (nx), dS y = dS cos (ny), dS z = dS cos (nz). 



Inserting these in the equation 84), dividing through by dS, 

 and taking the limit, as the edges of the tetrahedron become infinitely 

 small the ratio of the volume to the surface disappears, so that we 

 have finally 



86) X n = X x cos (nx) -f X. y cos (ny) -f X z cos (nz) , 

 and similarly 



Y n = Y x cos (nx) -\- Y y cos (ny) -f Y, cos (nz), 

 Z n = Z x cos (nx) -f- Zy cos (ny) -f Z z cos (nz). 



Let us now consider the equilibrium of any portion of the body 

 bounded by a closed surface S. Resolving in the X- direction, we 

 have as the condition for equilibrium, considering both the stresses 

 on the surface and the volume -forces, 



87) 

 Making use of equations 86) for X n , 



88) / / { X x cos (nx) + X y cos (ny) + X z cos (ne)} dS 



and by the divergence theorem, n being the outward normal, 



r r ridx sx ax 



89) JJJ y + w + ^ 



Since this must hold for every portion of the substance which is in 

 equilibrium, the integrand must vanish, and we have consequently 

 together with the result of resolving in the two other directions, 



dX dX dX 



These are but three of the six equations for equilibrium. The other 

 three are obtained by taking moments, the first being 



91) 



